This new set is due Monday, March 11, at the beginning of lecture.

Write a program that, given a square matrix , computes approximations to its eigenvalues using the QR-algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the error within which the approximations will be found. Apply your method to a matrix, and check the number of iterations the process requires.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. For this, you can work in groups of two or three. In case you cannot find anybody to work with, and do not know ow to program, let me know as soon as possible, and we will find an alternative.

(As extra credit problem, write a program for Francis’s algorithm as well, together with an explanation of your code, and apply the algorithm to the same matrix.)

For the remaining problems, you should turn in your own work. You can still collaborate with others, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc:

Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)

Let be an matrix with complex entries. Consider it as a linear transformation . Verify explicitly that is the unique matrix with the property that for all vectors . Recall that for and , their dot product is given by .

Check that if , then the eigenvalues of are real. To see this, let be an eigenvector of , and consider . In particular, this means that the eigenvalues of a symmetric matrix are real.

More significantly, for any Hermitian matrix , that is, one that satisfies , there is a basis consisting of eigenvectors of . We will prove this in due time.

43.614000-116.202000

Like this:

LikeLoading...

Related

This entry was posted on Thursday, February 28th, 2013 at 3:02 pm and is filed under 403/503: Linear Algebra II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

Ignas: It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here. […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]

Given any field automorphism of $\mathbb C$, the rational numbers are fixed. In fact, any number that is explicitly definable in $\mathbb C$ (in the first order language of fields) is fixed. (Actually, this means that we can only ensure that the rationals are fixed, I expand on this below.) Any construction of a wild automorphism uses the axiom of choice. Se […]

The Milner-Rado paradox is only a paradox in the traditional sense of the word: there are no inconsistencies here, but rather the result is (perhaps naively) seen as counter-intuitive. There are two reasons here: First, if $\alpha=\bigcup_{n

The question immediately reminded me of this. Here is an argument following the same basic idea at the beginning of that argument: First, consider $B=\{x\in A\mid A\cap(-\infty,x]$ is countable$\}$, and note that $B$ itself is countable: The point is that if $x\in B$ then $A\cap(-\infty,x]\subseteq B$. Now, if $B\ne\emptyset$, let $t=\sup B$, fix an increasi […]